Ductile

The /FAIL/BIQUAD, /FAIL/JOHNSON, and /FAIL/TAB1 failure models define material failure by relating the plastic strain at failure to the stress state in the material.

These failure models are often used to describe the ductile failure of materials. The state of stress in the material can be defined by using stress triaxiality.

Stress Triaxiality (Normalized Mean Stress)

For ductile materials, the state of stress (compression, shear, tension, etc.) of the material affects the plastic strain value at which the material will fail. An important and useful characteristic to describe the state of stress, stress triaxiality is defined as:(1)
σ * = σ m σ V M
Where,
σ m = 1 3 ( σ 1 + σ 2 + σ 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Wd8aadaWgaaWcbaWdbiaad2gaa8aabeaak8qacqGH9aqpdaWc aaWdaeaapeGaaGymaaWdaeaapeGaaG4maaaadaqadaWdaeaapeGaae 4Wd8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGHRaWkcaqGdpWd amaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiabgUcaRiaabo8apaWaaS baaSqaa8qacaaIZaaapaqabaaak8qacaGLOaGaayzkaaaaaa@465E@
Mean (hydrostatic) stress
σ V M = 1 2 [ ( σ 1 σ 2 ) 2 + ( σ 2 σ 3 ) 2 + ( σ 3 σ 1 ) 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Wd8aadaWgaaWcbaWdbiaadAfacaWGnbaapaqabaGcpeGaeyyp a0ZaaOaaa8aabaWdbmaalaaapaqaa8qacaaIXaaapaqaa8qacaaIYa aaamaadmaapaqaa8qadaqadaWdaeaapeGaae4Wd8aadaWgaaWcbaWd biaaigdaa8aabeaak8qacqGHsislcaqGdpWdamaaBaaaleaapeGaaG OmaaWdaeqaaaGcpeGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaaI YaaaaOGaey4kaSYaaeWaa8aabaWdbiaabo8apaWaaSbaaSqaa8qaca aIYaaapaqabaGcpeGaeyOeI0Iaae4Wd8aadaWgaaWcbaWdbiaaioda a8aabeaaaOWdbiaawIcacaGLPaaapaWaaWbaaSqabeaapeGaaGOmaa aakiabgUcaRmaabmaapaqaa8qacaqGdpWdamaaBaaaleaapeGaaG4m aaWdaeqaaOWdbiabgkHiTiaabo8apaWaaSbaaSqaa8qacaaIXaaapa qabaaak8qacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiaaikdaaaaa kiaawUfacaGLDbaaaSqabaaaaa@5A1B@
von Mises stress
Triaxiality values for some commons stress states can be derived as:
  • In pure tension:

    σ 2 = σ 3 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiaabo8apaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyypa0Ja ae4Wd8aadaWgaaWcbaWdbiaaiodaa8aabeaak8qacqGH9aqpcaaIWa aaaa@3E38@ , then σ * = σ m σ V M = 1 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaCaaaleqabaWdbiaacQcaaaGccqGH9aqpdaWcaaWd aeaapeGaae4Wd8aadaWgaaWcbaWdbiaad2gaa8aabeaaaOqaa8qacq aHdpWCpaWaaSbaaSqaa8qacaWGwbGaamytaaWdaeqaaaaak8qacqGH 9aqpdaWcaaqaaiaaigdaaeaacaaIZaaaaaaa@4329@

  • In biaxial compression:

    σ 1 = σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Wd8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH9aqpcaqG dpWdamaaBaaaleaapeGaaGOmaaWdaeqaaaaa@3BE9@ , and σ 3 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Wd8aadaWgaaWcbaWdbiaaiodaa8aabeaak8qacqGH9aqpcaaI Waaaaa@3A46@ , then σ * = σ m σ V M = 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaCaaaleqabaWdbiaacQcaaaGccqGH9aqpdaWcaaWd aeaapeGaae4Wd8aadaWgaaWcbaWdbiaad2gaa8aabeaaaOqaa8qacq aHdpWCpaWaaSbaaSqaa8qacaWGwbGaamytaaWdaeqaaaaak8qacqGH 9aqpcqGHsisldaWcaaqaaiaaikdaaeaacaaIZaaaaaaa@4417@

Stress triaxiality for various stress states:
Stress Triaxiality σ * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaCaaaleqabaWdbiaacQcaaaaaaa@38C9@
Stress State
2 3
Biaxial compression
1 3
Uniaxial compression
0
Pure shear
1 3
Uniaxial tension
1 3
Plain strain
2 3
Biaxial tension